=Paper=
{{Paper
|id=Vol-3348/short2
|storemode=property
|title=Mathematical Modeling of Steady Flow Distribution in Water Supply Networks with Pumping Stations and Regulating Capacitances
|pdfUrl=https://ceur-ws.org/Vol-3348/short2.pdf
|volume=Vol-3348
|authors=Sergey Dyadun,Sergey Yakovlev,Oleg Kobylin
|dblpUrl=https://dblp.org/rec/conf/profitai/DyadunYK22
}}
==Mathematical Modeling of Steady Flow Distribution in Water Supply Networks with Pumping Stations and Regulating Capacitances==
https://ceur-ws.org/Vol-3348/short2.pdf
Mathematical Modeling of Steady Flow Distribution in Water
Supply Networks with Pumping Stations and Regulating
Capacitances
Sergey Dyaduna, Sergey Yakovlevb,c and Oleg Kobylind
a
V.N. Karazin Kharkiv National University, Majdan Svobody 4, Kharkiv, 61077 Ukraine
b
Łódź University, ul. Prez. Gabriela Narutowicza 68, Łódź, 90-136 Poland
c
National Aerospace University “Kharkiv Aviation Institute”, Chkalov St. 17, Kharkiv, 61000, Ukraine
d
Kharkiv National University of Radio Electronics, 14 Nauki Ave, Kharkiv, 61166, Ukraine
Abstract
The article presents a mathematical model of the water supply system together with pumping
stations and regulating capacitances. This mathematical model is necessary to analyze the
quality of functioning of water supply systems when implementing control actions at
pumping stations, to evaluate the effectiveness of solving the problem of operational
planning of operating modes of water supply systems at a given control interval, as well as to
verify the correctness of decisions made on the control of technological processes of water
supply and distribution. The mathematical model provides the possibility of parallel
connection of an arbitrary number of pumping station units without the need for preliminary
equivalenting of their characteristics. Also, the developed mathematical model of the water
supply system, together with pumping stations and regulating capacitances, is extremely
important in simulation modeling of operational control systems for its operation modes.
Researchs was carried out using the method of imitation modeling of water supply systems
functioning of many major cities.
Keywords 1
mathematical modeling, water supply system, functioning, steady flow distribution, pumping
station, regulating capacitance, control.
1. Introduction
The purpose of this article is to develop a mathematical model of the water supply system together
with pumping stations and regulating capacitances. In the control of water supply systems (WSS) and
simulation modeling of operational control systems for their operation modes, an important role is
played by the task of analyzing the steady distribution of flows in networks. To solve this problem, it
is necessary to have a mathematical model of the water supply system, including pumping stations
(PS) operating for it and regulating capacitances, which would take into account the possibility of
parallel connection of a different number of pumping station units without the need of preliminary
equivalenting of their characteristics. Such a model is necessary to analyze the quality of functioning
of water supply systems when implementing control actions at pumping stations, to evaluate the
effectiveness of solving the problem of operational planning of operating modes of water supply
systems at a given control interval, as well as for verify the correctness of decisions made on the
control of water supply and distribution technological processes. Also, the mathematical model of the
water supply system, together with pumping stations and regulating capacitances, is extremely
important in simulation modeling of operational control systems for its operation modes.
2nd International Workshop of IT-professionals on Artificial Intelligence (ProfIT AI 2022), December 2-4, 2022, Łódź, Poland
EMAIL: daulding@ukr.net (S. Dyadun); svsyak7@gmail.com (S. Yakovlev); oleg.kobylin@nure.ua (O. Kobylin)
ORCID: 0000-0002-1910-8594 (S. Dyadun); 0000-0003-1707-843X (S. Yakovlev); 0000-0003-0834-0475 (O. Kobylin)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
2. Analysis of Literature Data and Statement of the Problem
Many fundamental scientific works of Abramov N.N., Evdokimov A.G., Tevyashev A.D., Hasilev
V.Y., Merenkov V.P., Novitsky N.N. and other scientists [3] in the 70s of the last century are devoted
to the development of mathematical models of water supply networks. Basically, they were
considered to solve the problem of hydraulic calculation of their operating modes, which is very
important in the design of water supply networks.
The purpose of this task was to calculate the values of flow rates in all arcs of the water supply
network graph and pressures in all its nodes with a certain combination of initial data that would
provide pressures in the nodes with consumers connected to them not lower than some given
minimum allowable value determined by the height of buildings and geodetic relief terrain.
In the works of Evdokimov A.G. [1, 2] also considered the problem inverse to this one - the problem
of analyzing the steady flow distribution in the network, i.e. determining the values of flow rates in all
arcs of the water supply network graph and pressures in all its nodes for fixed values of flow rates and
pressures at the network inlets, i.e. at the outlets of pumping stations. As a result of solving this
problem, the water in the network seemed to be distributed by gravity, and in the nodes with
consumers connected to them, the pressure could be either above the minimum allowable value or
below it. That is, in the second case, some or all consumers connected to such nodes received less
water (on the upper floors of houses or high-lying areas), or did not receive it at all.
This task is extremely important in the operation of water supply networks, for dispatchers, i.e.
persons making decisions on the control of the technological process of water supply. For dispatchers
of water supply systems this were not very comfortable and did not manage to quickly provide the
necessary values of flow rates and pressures at the outlets of all pumping stations, i.e. at the inlets of
the water supply networks. Talking with the chief dispatchers and directors of the water supply
companies of many large cities, it became clear their desire to see a complete picture of the constantly
changing situations in the city's water supply network. And for this, it is necessary to know every time
how the water flows will be distributed, and how consumers will be provided in the network, with
various combinations of switched on pumping units, with known water levels in the reservoirs of
pumping stations, and with smooth regulation of the supply of pumping units using asynchronous
valve cascades.
Therefore, a mathematical model of the water supply system was developed together with
pumping stations and regulating capacitances. This mathematical model provides the possibility of
parallel connection of an arbitrary number of pumping station units without the need for preliminary
equivalenting of their characteristics. Such a mathematical model is necessary to analyze the quality
of functioning of water supply systems when implementing control actions at pumping stations, to
evaluate the effectiveness of solving the problem of operational planning of operating modes of water
supply systems at a given control interval, as well as to verify the correctness of decisions made on
the control of technological processes of supply and distribution of water.
Also, the developed mathematical model of the water supply system, together with pumping
stations and regulating capacitances, is extremely important in simulation modeling of operational
control systems for its operation modes.
This mathematical model is presented in this article.
3. Research results
Let G (V, E ) is a graph of the WSS that models its structure and displays the relationships
between individual elements. Here V is the set of all WSS nodes; E is the set of all arcs of the WSS.
Let's connect all the inputs and outputs of the graph G (V, E ) , through which water enters the
network and is taken from it, respectively, with a zero fictitious point.
Let L be the set of pumping stations and regulating capacitances, then the elements of this set will
be fictitious arcs connecting the zero point with the inputs of all PS and regulating capacitances; M is
the set of passive elements, i.e. main sections (pipelines) of the water supply network, which are real
arcs of the graph; N is the set of WSS nodes with consumers connected to them, i.e. the set of
fictitious arcs of the WSS model. At the same time, the set of nodes of the WSS graph consists of two
non-intersecting subsets V = N N , where N is a subset of intermediate nodes of the WSS
model, i.e. such nodes in which there is no water intake, however they need to be reflected in the
WSS model. In addition, we introduce a set K =
L j which characterizing the total number of arcs
jL
with pumping units at all pumping stations of the WSS. Thus, the elements of the set are the links of
all pumping stations, each of which directly includes the active element itself (pump), as well as
sections adjacent to it with adjustable and unregulated valves. At the same time, as is known [3], each
link of the pumping station corresponds to an equation that describes the process of water movement
through this pumping unit:
H IN - ri1 q i2 + ( Ψ0i + Ψ1i qi + Ψ 2i q i2 ) ( Di /Di ) ( n i /ni ) -
2 2
(1)
- ri3 ( λi ) qi2 - H OUT = 0, i L j .
Here HIN , HOUT is the pressure at the inlet and outlet of the PS, respectively; q i - water flow
through the i-th link; 0i , 1i , 2i - coefficients of approximation of the load characteristics H(q )
of the i-th pumping unit; ri1 , ri3 - resistance of sections with unregulated and adjustable gate valves
located in the suction and pressure lines of the i-th pump, respectively; i - the degree of opening of
the i-th adjustable gate valve; D i , Di - respectively, the normal and cut impeller diameter of the
i-th pump; n i , n i - respectively, the actual and nominal speed of the impeller of the i-th pumping unit
(if it has an adjustable drive).
For the unambiguity of the direction of the flow of water pumped by the i-th switched on WSS
unit, we represent (1) in a slightly different form:
H IN - ri1 q i |q i | + ( Ψ0i + Ψ1i |qi | + Ψ 2i q i |q i | ) ( Di /Di ) ( n i /n i ) -
2 2
(2)
- ri3 ( λi ) qi |q i | - H OUT = 0, i L j.
The pressure loss for an active element, whether it is a PS or a separately operating pumping unit,
is always a negative value, so in this case we can talk about the “acquisition” of pressure.
Let's choose a tree of the WSS graph in such a way that it includes the magistral sections of the
network and sections with pumps (belonging to different PS of the WSS), as well as one fictitious
branch connecting the zero point with the input of some of the PS.
Let us assign number 1 to it. In this case, all fictitious sections incident to the WSS nodes, which
are the inputs of the PS (except for the first one) and regulating capacitances, as well as sections with
consumers, will be assigned to chords, while the magistral sections and sections with pumps will
partially become chords, and partially - the branches of a tree. We believe that index 1, assigned by
sets, respectively L, M, N, K, E , characterizes the belonging of their elements to the branches of the
tree, and index 2, to chords. As a result of such a choice, the set of all arcs of the WSS graph can be
represented as E = E1 E 2 , where E1 = M1 K1 L1 , E 2 = M 2 K 2 L 2 N 2 , L1 = 1,
N1 = , N 2 = N.
(a ) (p) (a )
The set L 2 is divided into two non-intersecting subsets L 2 = L 2 L 2 , where L 2 is the set of
(p)
chords with active elements (PS); L 2 - a set of chords with regulating capacitances (water towers,
columns, reservoirs, which are passive-active control elements of the network).
In addition, we denote q i - the water flow in the i-th section of the network, i M; ri - the
(г )
hydraulic resistance of the i-th section of the network, i M; h i - the difference in geodetic marks
of the beginning and end of the i-th section of the network,; h i - pressure loss in the i-th section of the
network, i M.
Taking into account the choice of the tree of the graph of the WSS, as well as the fact that the sum
of the differences in geodetic heights for any closed cycle containing the backbone sections of the
network is equal to zero, i.e.
h i( г ) + b1ri h (r r ) = 0, i M 2 , (3)
rM1
the mathematical model of the steady flow distribution in the water supply network, together with
active sources and regulating capacitances, will take the form
f r = sign q r rr | q r | 2 + b1ri sign q i ri | q i | 2 = 0, r M 2 ; (4)
iM1
f r = H IN1 + Ψ0k + Ψ1k |q k | + Ψ2k q k |q k | -h r - b1ri ( signq i ri |q i |2 + h i(г) ) = 0,
iM1 (5)
r N, k K1;
f k = 0 k + 1k | q k | + 2 k q k | q k | + x i (0i + 1i | q i | + 2i q i | q i | ) = 0,
(6)
k K 2 , i K1 ;
( )
f r = sign q r rr | q r | 2 + h (rr ) + h (r p ) + b1ri sign q i ri | q i | 2 + h i(г ) + 0 k + 1k | q k | +
iM1 (7)
+ 2 k q k | q k | = 0, r L(2p ) , k K1 ;
f r = H IN1 - H INr + signq k ( Ψ 0k + Ψ1k |q k | + Ψ2k q k |q k | ) -
kK1
(8)
- b1ri ( signq i ri |q i |2 + h i(г) ) = 0, r L(a)2 ;
iM1
q i = b1ri q r + x k q k + Q i+ , i M1 L1 K1 , (9)
rM 2 kK 2
where 2 k = 2 k (D k / Dk ) (n k / n k ) − rk1 − rk 3 ( k ),
2 2
k K1 K 2 ,
1, if the i-th pump is on,
xi = 0, if the i-th pump is off, i K1 K1 ;
Q i+ = b1ki q k = const;
kN L 2
H IN1 , H INk are the inlet pressure of the 1st and k-th PS, respectively;
b1ri is an element of the cyclomatic matrix В1 [1-2].
The value marked with the index "+" is considered to be set.
In the above mathematical model of the WSS, it is assumed that the fictitious sections with
consumers are directed from the network to the zero fictitious point, and the sections with pumps and
the fictitious sections corresponding to them are vice versa.
Let's analyze the conditions for the solvability of the system of equations of the mathematical
model of the water supply network together with active sources and regulating capacitances. It can be
solved if the boundary conditions for the functioning of the WSS are given in the form of a
combination of values of variable flow rates and pressures at its inlets and outlets.
4. Conclusions
A mathematical model of the steady flow distribution in water supply systems containing
pumping stations and regulating capacitances is necessary to analyze the quality of functioning of
water supply systems when implementing control actions at pumping stations [2; 8; 9], to evaluate the
effectiveness of solving the problem of operational planning of operating modes of water supply
systems at a given control interval [2; 4; 5, 6; 7], as well as for verify the correctness of decisions
made on the control of water supply and distribution technological processes [2; 4; 5, 6; 7].
A simulation model of the technological processes of water supply systems functioning together
with pumping stations and regulating capacitances on any given time interval has been developed.
It is based on the solution of the problem of analyzing the steady flow distribution in the water
supply network together with pumping stations and control tanks, which is determined by solving the
system of equations [1, 2, 6, 7] of the corresponding mathematical model when setting a combination
of flow rates and pressures as boundary conditions at the inlets and outlets of the water supply
systems.
Researchs was carried out using the method of imitation modeling of water supply systems
functioning of many major cities. Currently, all software is implemented in C++. In some cases, in
addition to own software, individual EPANET functions can be used [10]. The developed databases
were implemented with MS Access DBMS.
The use and widespread adoption of information technologies of optimal control operation of
WSS allows in practice to improve the quality and effectiveness of their functioning by reducing the
excess pressure in the networks (and, consequently, reducing unproductive costs of water), reducing
electricity costs, reducing the probability of occurrence of emergency situations in networks.
5. References
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[2] Evdokimov A.G. Optimal problems on the engineering networks. - Kharkov: Higher School,
153 p. [in Russian] (1976)
[3] Fallside F., Perry P.F., Burch R.H., Marlow K.C. The Development of Modelling and
Simulation Techniques Applied to a Computer - Based - Telecontrol Water Supply System. In:
Computer Simulation of Water Recourses Systems, No.12, pp. 617-639 (1975)
[4] Tevyashev A.D., Matvienko O.I. About one approach to solve the problem of management of the
development and operation of centralized water– supply systems. In: Econtechmod. An
International Quarterly Journal. Vol. 3, Issue 3, pp. 61–76 (2014)
[5] Tevyashev A.D., Matvienko O.I. The mathematical model and the method of optimal stochastic
control over the modes of the water main operation // Eastern-European Journal of Enterprise
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[6] Dyadun S.V., Kuznetsov V.N., Esilevskiy V.S. Comparative analysis of efficiency of methods of
optimizations of flows distribution in water supply systems with large number of active sources //
”Mathematical Models and Methods of the Analysis and Optimal Synthesis of the Developing
Pipeline and Hydraulic Systems”, Section: Control of Functioning of Pipeline System, 10p.
(2020)
[7] Sergey Dyadun. Information Technologies to Estimation the Effectiveness of Water Supply
Systems Control Depending on the Degree of Model Uncertainty // ”ICT in Education, Research,
and Industrial Applications: Integration, Harmonization, and Knowledge Transfer” – ICTERI
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[8] Pulido–Calvo, I. Gutiérrez–Estrada, J.C.: Selection and operation of pumping stations of water
distribution systems. In: Environmental Research Journal, Nova Science Publishers. Vol. 5,
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stations. In: Florida Water Resources Journal, pp. 44–52 (2011)
[10] EPANET: Application for Modeling Drinking Water Distribution Systems. Available at: https
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